Figure 2.

(a) Double-log histogram showing the distribution of binned step lengths from a four-level BM process, observed at different temporal resolutions (lags). (b) A truncated Lévy walk shows the expected linear slope in a log–log transformation of the step-length distribution, but the scale-free pattern beaks down for step-length bins larger than the truncation scale. (c) The step-length pattern from a truncated multi-scaled random walk (truncated LW-like, but extended with memory influence and return steps) shape-shifts from power-law compliance (t_obs = 1), through a ‘hockey stick’ pattern (t_obs = 10) to a truncated power-law pattern (larger t_obs). (d) A multi-scaled random walk without step-length truncation shows a relatively noisy step-length frequency over the extreme step-length bins. Two independent distributions, dashed and solid lines, are included for the two observational scales t_obs = 100 (blue-coloured) and t_obs = 1000 (red-coloured), to illustrate how the frequency peaks over the noisy range vary stochastically between series when t_obs is similar to—or larger—than the temporal scale for return steps (in these iterations, a return event at frequency 1 : 100 on average). The reason for the noisiness at these coarse levels of observations is explained in §4.